{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:LTJ3BGKIKIWUAGDFEPQKHTYOSR","short_pith_number":"pith:LTJ3BGKI","schema_version":"1.0","canonical_sha256":"5cd3b09948522d40186523e0a3cf0e946aa83f2e508963ece7ae0ff9e2d24ff9","source":{"kind":"arxiv","id":"1711.08235","version":4},"attestation_state":"computed","paper":{"title":"A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ralf Zimmermann","submitted_at":"2017-11-22T11:33:07Z","abstract_excerpt":"In this work, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given matrix $X\\in \\mathbb{R}^{n\\times p}$ with known matrix factorization $X = UW$, where $U\\in\\mathbb{R}^{n\\times p}$ is column-orthogonal, i.e. $U^TU=I$. Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U\\Sigma V^T$, and the QR-decomposition, where $X = UW = QR$. An elementary approach to produce a column-orthogonal matrix $U_{new}$, whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$ is via applying a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1711.08235","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-11-22T11:33:07Z","cross_cats_sorted":[],"title_canon_sha256":"4a7db15dc4c631c857ba54d1203b6694e26da8c81c9218915bad7ae7c61a3495","abstract_canon_sha256":"df1fe5fbf2b8bbac1a1ff34a47afb447a9ecc446f0f4a81d72a759a95123b05a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:08.222051Z","signature_b64":"Jc3hQ0aOWN4917WdoNhyu5TfGcOr9qtX/kbbP6uW8mr4UkHDbdHM8qUSm0mP0YJp4HqvSmKR5zQEkcl2zpuuCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cd3b09948522d40186523e0a3cf0e946aa83f2e508963ece7ae0ff9e2d24ff9","last_reissued_at":"2026-05-17T23:52:08.221566Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:08.221566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ralf Zimmermann","submitted_at":"2017-11-22T11:33:07Z","abstract_excerpt":"In this work, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given matrix $X\\in \\mathbb{R}^{n\\times p}$ with known matrix factorization $X = UW$, where $U\\in\\mathbb{R}^{n\\times p}$ is column-orthogonal, i.e. $U^TU=I$. Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U\\Sigma V^T$, and the QR-decomposition, where $X = UW = QR$. An elementary approach to produce a column-orthogonal matrix $U_{new}$, whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$ is via applying a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08235","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1711.08235","created_at":"2026-05-17T23:52:08.221646+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.08235v4","created_at":"2026-05-17T23:52:08.221646+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.08235","created_at":"2026-05-17T23:52:08.221646+00:00"},{"alias_kind":"pith_short_12","alias_value":"LTJ3BGKIKIWU","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"LTJ3BGKIKIWUAGDF","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"LTJ3BGKI","created_at":"2026-05-18T12:31:28.150371+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR","json":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR.json","graph_json":"https://pith.science/api/pith-number/LTJ3BGKIKIWUAGDFEPQKHTYOSR/graph.json","events_json":"https://pith.science/api/pith-number/LTJ3BGKIKIWUAGDFEPQKHTYOSR/events.json","paper":"https://pith.science/paper/LTJ3BGKI"},"agent_actions":{"view_html":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR","download_json":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR.json","view_paper":"https://pith.science/paper/LTJ3BGKI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.08235&json=true","fetch_graph":"https://pith.science/api/pith-number/LTJ3BGKIKIWUAGDFEPQKHTYOSR/graph.json","fetch_events":"https://pith.science/api/pith-number/LTJ3BGKIKIWUAGDFEPQKHTYOSR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR/action/storage_attestation","attest_author":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR/action/author_attestation","sign_citation":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR/action/citation_signature","submit_replication":"https://pith.science/pith/LTJ3BGKIKIWUAGDFEPQKHTYOSR/action/replication_record"}},"created_at":"2026-05-17T23:52:08.221646+00:00","updated_at":"2026-05-17T23:52:08.221646+00:00"}